Al-Khwarizmi

• 1. Introduction

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  Al-Khwarizmi gave the world two words it still uses every day, and most people who say them have no idea. The word algebra comes from the title of one of his books. The word algorithm comes from the Latinized form of his own name. He was born around 780 and worked during the Islamic Golden Age, producing works in mathematics, astronomy, and geography written entirely in Arabic. Around 820 he was at the House of Wisdom in Baghdad, the capital of the Abbasid Caliphate. Yet for a man whose ideas shaped how Europe would count and calculate, almost nothing about his life is known with certainty. Who was he, and where did he come from? How did a treatise compiled in Baghdad become the principal mathematics textbook of European universities for centuries? And why do the words digit, algorism, and algorithm all trace back to a single scholar?

• 2. A Life In Shadow

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  Ibn al-Nadim, whose Al-Fihrist includes a short biography and a list of his books, gives his birthplace as Khwarazm. His name means 'from Khwarazm', a region that was part of Greater Iran and is now part of Turkmenistan and Uzbekistan. He was of Persian stock, probably Khwarezmian, yet he wrote his scientific works entirely in Arabic despite his non-Arabic origin. Al-Tabari recorded his name as Muhammad ibn Musa al-Khwarizmi al-Majusi al-Qutrubbulli, and that string of epithets has caused centuries of confusion. The epithet al-Qutrubbulli could suggest he came from Qutrubbul, near Baghdad. David A. King affirms this nisba, noting he was called al-Qutrubbulli because he was born just outside of Baghdad. Roshdi Rashed disagrees entirely, arguing the citation should be read as naming two separate people, with the Arabic word for 'and' dropped in an early copy. The epithet al-Majusi seemed to mark him as an adherent of the old Zoroastrian religion. But Toomer points to the pious preface of his Algebra as proof he was an orthodox Muslim. Toomer suggests the epithet meant only that his forebears, and perhaps he in his youth, had been Zoroastrians. Even his identity is contested. Douglas Morton Dunlop suggested he might have been the same person as Muhammad ibn Musa ibn Shakir, eldest of the three Banu Musa brothers.

• 3. The House Of Wisdom

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  Around 820 CE, al-Khwarizmi was appointed astronomer and head of the library of the House of Wisdom, an institution established by the Abbasid Caliph al-Ma'mun. After the Muslim conquest of Persia, Baghdad had become the centre of scientific studies and trade. He accomplished most of his work between 813 and 833, studying sciences and mathematics and translating Greek and Sanskrit scientific manuscripts. Beyond mathematics, he was also a historian, cited by figures such as al-Tabari and Ibn Abi Tahir. His reach extended into diplomacy as well. During the reign of al-Wathiq, he is said to have been involved in the first of two embassies to the Khazars. The patronage of al-Ma'mun did more than fund a library. It encouraged the very book that would carry his name into the languages of Europe.

• 4. The Birth Of Algebra

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  Al-Jabr, whose full title translates as The Compendious Book on Calculation by Completion and Balancing, was written approximately 820 CE with the encouragement of Caliph al-Ma'mun. It was a popular work, replete with examples drawn from trade, surveying, and legal inheritance. It presented the first systematic solution of linear and quadratic equations, and showed how to solve quadratic equations by completing the square, with geometric justifications. The word algebra comes from al-jabr, meaning restoration, the act of adding a number to both sides of an equation to consolidate or cancel terms. His method reduced every equation to one of six standard forms, then applied two operations: al-jabr, the restoring, and al-muqabala, the balancing. Al-jabr removes negative units, roots, and squares by adding the same quantity to each side. Al-muqabala brings quantities of the same type to the same side. In his day most modern notation had not yet been invented, so he worked in ordinary text. Even numbers were written out in words rather than symbols. Solomon Gandz called him more entitled to be the father of algebra than Diophantus, because he was the first to teach algebra in an elementary form and for its own sake. Victor J. Katz noted his work, written in Baghdad around 825, is the first true algebra text still extant. Roshdi Rashed and Angela Armstrong observed that his text introduces the idea of an equation for its own sake from the very beginning. A unique Arabic copy is kept at Oxford and was translated in 1831 by F. Rosen.

• 5. Numerals For The West

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  On the Calculation with Hindu Numerals, written about 825, spread the Hindu-Arabic numeral system throughout the Middle East and Europe. His arithmetic writings include kitab al-hisab al-hindi, the Book of Indian computation, and perhaps a more elementary text on addition and subtraction in Indian arithmetic. These texts described algorithms on decimal numbers that could be carried out on a dust board, called takht in Arabic. A board covered with a thin layer of dust or sand let figures be written with a stylus and easily erased. His algorithms were used for almost three centuries, until replaced by Al-Uqlidisi's algorithms that worked with pen and paper. The original Arabic of this work is lost, surviving only in Latin translations. Four Latin adaptations survive, none believed to be a literal translation, including Dixit Algorizmi, meaning 'Thus spake Al-Khwarizmi'. That manuscript sits in the University of Cambridge library, attributed to Adelard of Bath, who had translated the astronomical tables in 1126. His Latinized name, Algorismus, became the name of the method itself, gradually replacing the abacus-based methods used in Europe. Both 'algorithm' and 'algorism' descend from the Latinized forms of his name, Algoritmi and Algorismi.

• 6. Mapping The Stars

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  His Zij as-Sindhind, the astronomical tables of Siddhanta, contains roughly 37 chapters and 116 tables of calendrical, astronomical, and astrological data, plus a table of sine values. The word Sindhind is a corruption of the Sanskrit Siddhanta, the usual designation of an astronomical textbook. The mean motions in his tables derive from the corrected Brahmasiddhanta of Brahmagupta. The tables track the movements of the Sun, Moon, and the five planets known at the time. The original Arabic, written about 820, is lost. A version by the Spanish astronomer Maslama al-Majriti, made around 1000, survives in a Latin translation. Four surviving manuscripts of that Latin translation are kept at the Bibliotheque publique in Chartres, the Bibliotheque Mazarine in Paris, the Biblioteca Nacional in Madrid, and the Bodleian Library in Oxford. His astronomical tables became essential tools for later generations, enabling calculations to predict eclipses and the positions of celestial bodies, crucial for navigation and timekeeping. The same Zij carried tables for the trigonometric functions of sine and cosine, and a related treatise on spherical trigonometry is attributed to him. He produced accurate sine and cosine tables.

• 7. Redrawing The Earth

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  Kitab Surat al-Ard, the Book of the Description of the Earth, was finished in 833 as a major reworking of Ptolemy's second-century Geography. It lists 2402 coordinates of cities and other geographical features following a general introduction. One surviving copy is kept at the Strasbourg University Library, with a Latin translation at the Biblioteca Nacional de Espana in Madrid. He assisted a project to determine the circumference of the Earth and to make a world map for the caliph al-Ma'mun, overseeing 70 geographers. He corrected Ptolemy's gross overestimate for the length of the Mediterranean Sea from the Canary Islands to its eastern shores. Ptolemy had estimated it at 63 degrees of longitude; al-Khwarizmi almost correctly estimated nearly 50 degrees. He depicted the Atlantic and Indian Oceans as open bodies of water, not the land-locked seas Ptolemy had drawn. Neither the Arabic copy nor the Latin translation includes the map of the world. Hubert Daunicht reconstructed the missing map from the list of coordinates, reading the latitudes and longitudes of coastal points, transferring them onto graph paper, and connecting them with straight lines. His Prime Meridian at the Fortunate Isles sat around 10 degrees east of the line used by Marinus and Ptolemy. Most medieval Muslim gazetteers continued to use his prime meridian.

• 8. The Long Afterlife

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  Robert of Chester translated Al-Jabr into Latin at Segovia in 1145, under the title Liber algebrae et almucabala, and Gerard of Cremona produced his own version. That book was used until the 16th century as the principal mathematical textbook of European universities. The reach of his work was not limited to algebra and arithmetic. He wrote a treatise on the Hebrew calendar describing the Metonic cycle, a 19-year intercalation cycle, and rules for determining the day of the week on which Tishrei begins. Ibn al-Nadim's Al-Fihrist mentions his Kitab al-Tarikh, a book of annals; no direct manuscript survives, but a copy had reached Nusaybin by the 11th century, where the metropolitan bishop Mar Elias bar Shinaya found it and quoted it. Scattered manuscripts in Berlin, Istanbul, Tashkent, Cairo, and Paris hold further material likely his, including a paper on sundials and one on determining the direction of Mecca. He wrote two books on using and constructing astrolabes. His name now circles places he never saw: a crater on the far side of the Moon bears his name, and so do two main-belt asteroids, one discovered in 1986 and one in 1997.